IMPROVEMENT OF THE TECHNIQUE FOR CALCULATING THE DEFLECTION OF A SINGLE-INCLINED BEAM WITH VARIABLE RIGIDITY ALONG LENGTH

Abstract

The article discusses the solution of the theoretical problem of determining the deflection of a single-inclined beam with a linear change in stiffness along the span. The urgency of solving such a problem is due to the need to ensure normal operating conditions and comply with safety conditions. The improvement of the method for determining the maximum deflections of beam elements is based on the fact that, according to the design standards of a reinforced concrete beam, the deflection must be determined according to the general rules of structural mechanics. The case is considered when the stress in the structure is much less than the limiting value. Then the plastic component of deformation is relatively small. The object of theoretical research is a single-span hingedly supported single-inclined beam of rectangular cross-section, loaded with a uniformly distributed linear load. Most steel and reinforced concrete beams have I-beams cross section, for which the axial moment of inertia in the bending plane is roughly proportional to the cube of the height. Therefore, for simplicity, a rectangular section is adopted. Based on the geometric scheme of the beam, a linear relationship was obtained between the coordinate along the span and its height. On this basis, the function of the axial moment of inertia of the cross section is compiled. To obtain an analytical formula for the deflections and angles of beam rotation along the length of the span, the differential equation of the curved axis was integrated. The bending moment in the beam section from a given linear load is presented as a quadratic dependence. Successive integration of the differential equation made it possible to obtain the functions of the angle of rotation and deflection. Integration constants are found on the assumption that the deflections on the left and right supports are equal to zero. For practical confirmation of the correctness of the result obtained for deflections, a special case was considered when the slope of the beam is equal to zero. Analysis of the deformation formula of the beam showed that it is necessary to disclose the mathematical uncertainty using the Lopital’s rule. This problem is associated with certain mathematical difficulties, and it was solved using the MathCAD computer environment. The problem of finding the deflections and angles of rotation of the beam was solved with the control initial data. Using the computer environment MathCAD, a graphical solution to the differential equation of the curved axis was directly obtained, and graphs of the functions of deflections and angles of rotation were also built. Analysis of these graphs showed that the maximum deflection and zero angle of rotation have the same abscissa, which corresponds to the theoretical assumptions. It was shown that the beam has a maximum deflection not in the middle of the span, but closer to the left support, where its height is less.